New Sequence Spaces Derived by the Composition of Binomial and Quadruple Band Matrices

Yıl 2024, Cilt: 5 Sayı: 2, 76 - 92, 31.12.2024

Kübra Topal

https://doi.org/10.53501/rteufemud.1478846

Öz

In this work, we construct the sequence spaces b_0^(a,b) (Q), b_c^(a,b) (Q) and b_∞^(a,b) (Q), derived from the combination of binomial and quadruple band matrices, where Q is a quadruple band matrix. The study is divided into four main sections. The first section provides some fundamental definitions and equations that will be used later. In the second section, detailed discussions are made on some works previously completed by various authors using the domain of the binomial matrix. Then, the three new sequence spaces b_0^(a,b) (Q), b_c^(a,b) (Q) and b_∞^(a,b) (Q) are defined. It is then shown that these obtained sequence spaces are BK-spaces. Following this, it is demonstrated that b_0^(a,b) (Q), b_c^(a,b) (Q) and b_∞^(a,b) (Q) sequence spaces are linearly isomorphic to the spaces c_0, c and l_∞, in turn, followed by some inclusion relations. In the third section, the Schauder bases of the new sequence spaces b_0^(a,b) (Q) and b_c^(a,b) (Q) are provided, and the α-, β- and γ- duals of b_0^(a,b) (Q), b_c^(a,b) (Q) and b_∞^(a,b) (Q) sequence spaces are determined. The fourth section characterizes some matrix classes. As a result, it is observed that the new matrix obtained from the combination of binomial and quadruple band matrices provides more general and comprehensive results than those obtained previously.

Anahtar Kelimeler

Matrix transformations, Matrix domain, Schauder basis, α- β- and γ- duals, Matrix classes

Binom ve Dörtlü Band Matrisinin Kompozisyonu ile Türetilmiş Yeni Dizi Uzayları

Öz

Bu çalışmada, Q dörtlü band matrisi olmak üzere, binom ve dörtlü band matrisinin birleşiminden türetilmiş olan b_0^(a,b) (Q), b_c^(a,b) (Q) ve b_∞^(a,b) (Q) dizi uzaylarını oluşturuyoruz. Çalışma dört ana bölüm olarak hazırlandı. İlk bölümde, daha sonra kullanacağımız bazı temel tanım ve eşitlikler verildi. İkinci bölümde, daha önce çeşitli yazarlar tarafından binom matrisinin etki alanı kullanılarak yapılmış çalışmalardan bazılarına ayrıntılı olarak değinildi. Ardından, üç yeni dizi uzayı olan b_0^(a,b) (Q), b_c^(a,b) (Q) ve b_∞^(a,b) (Q) tanımlandı. Daha sonra elde edilen bu dizi uzaylarının BK-uzayı olduğu gösterildi. Bundan sonra b_0^(a,b) (Q), b_c^(a,b) (Q) ve b_∞^(a,b) (Q) dizi uzaylarının sırasıyla c_0, c ve l_∞ dizi uzaylarına lineer olarak izomorf oldukları gösterildi. Ardından bazı içerme bağınları verildi. Üçüncü bölümde elde ettiğimiz yeni dizi uzaylarından b_0^(a,b) (Q) ve b_c^(a,b) (Q) nin Schauder bazları verildi ve b_0^(a,b) (Q), b_c^(a,b) (Q) ve b_∞^(a,b) (Q) dizi uzaylarının α-, β- ve γ- dualleri belirlendi. Dördüncü bölümde bazı matris sınıflarını karakterize edildi. Sonuç olarak binom ve dörtlü band matrisinin birleşiminden elde edilen yeni matrisin daha önce elde edilenlerden daha genel ve kapsamlı sonuçlar verdiği görüldü.

Anahtar Kelimeler

Matris dönüşümleri, Matris etki alanı, Schauder bazı, α- β- ve γ- dualleri, Matris sınıfları

Kaynakça

• Altay, B., Başar, F. (2005). On some Euler sequence spaces of nonabsolute type. Ukrainian Mathematical Journal, 57, 1–17. https://doi.org/10.1007/s11253-005-0168-9
• Altay, B., Başar, F., Mursaleen, M. (2006). On the Euler sequence spaces which include the spaces l_p and l_∞ I. Information Sciences, 176(10), 1450-1462. https://doi.org/10.1016/j.ins.2005.05.008
• Başar, F., Altay, B. (2003). On the space of sequences of p-bounded variation and related matrix mappings. Ukrainian Mathematical Journal, 55, 136–147. https://doi.org/10.1023/A:1025080820961
• Bilgin Ellidokuzoğlu, H., Demiriz, S. (2018). Some algebraic and topological properties of new Lucas difference sequence spaces. Turkish Journal of Mathematics and Computer Science, 10, 144-152.
• Bilgin Ellidokuzoğlu, H., Demiriz, S. (2021). [l_p ]_(e.r) Euler-Riesz difference sequence space. Analysis in Theory and Applications, 37(4), 557-571. https://doi.org/10.4208/ata.OA-2017-0068
• Bişgin, M.C. (2016a). The binomial sequence spaces of nonabsolute type. Journal of Inequalities and Applications, 309. https://doi.org/10.1186/s13660-016-1256-0
• Bişgin, M.C. (2016b). The binomial sequence spaces which include the spaces l_p and l_∞ and geometric properties. Journal of Inequalities and Applications, 304. https://doi.org/10.1186/s13660-016-1252-4
• Bişgin, M.C. (2017a). A note on the sequence space b_p^(r,s) (G). Cumhuriyet Science Journal, 38(4), 11-25. https://doi.org/10.17776/csj.363211
• Bişgin, M.C. (2017b). Binom ve üçlü band matrislerinin kompozisyonu ile türetilen bazı yeni dizi uzayları. 12. Ankara Matematik Günleri, 25 – 26 Mayıs 2017, Ankara, Türkiye.
• Bişgin, M.C. (2023). Four new sequence spaces obtained from the domain of quadruple band matrix operator. Turkish Journal of Mathematics and Computer Science, 15(2), 294-311. https://doi.org/10.47000/tjmcs.1274395
• Choudhary, B., Nanda, S. (1989). Functional Analysis with Applications. Wiley, ISBN: 047021564X, New Delhi, 344.
• Demiriz, S., Çakan, C. (2011). Some topological and geometrical properties of a new difference sequence space. Abstract and Applied Analysis, 2011, 213878. https://doi.org/10.1155/2011/213878
• Demiriz, S., Erdem, S. (2020a). On the new double binomial sequence space. Turkish Journal of Mathematics and Computer Science, 12(2), 101-111. https://doi.org/10.47000/tjmcs.727448
• Demiriz, S., Erdem, S. (2020b). Domain of binomial matrix in some spaces of double sequences. Punjab University Journal of Mathematics, 52(11), 65-79.
• Erdem, S., Demiriz, S. (2020). Almost convergence and 4-dimensional binomial matrix. Konuralp Journal of Mathematics. 8(2), 329-336.
• Erdem, S., Demiriz, S. (2021). A study on strongly almost convergent and strongly almost null binomial double sequence spaces. Fundamental Journal of Mathematics and Applications. 4(4), 271-279. https://dx.doi.org/10.33401/fujma.987981
• Et, M. (1993). On some difference sequence spaces. Turkish Journal of Mathematics, 17, 18-24.
• Kızmaz, H. (1981). On certain sequence spaces. Canadian Mathematical Bulletin, 24(2), 169-176. https://doi.org/10.4153/CMB-1981-027-5
• Meng, J., Song, M. (2017). On some binomial difference sequence spaces. Kyungpook Mathematical Journal, 57, 631-640. https://doi.org/10.5666/KMJ.2017.57.4.631
• Sönmez, A. (2019). Some new sequence spaces derived by the composition of binomial matrix and double band matrix. Journal of Applied Analysis and Computation, 9(1), 231-244. http://doi.org/10.11948/2019.231
• Sönmez, A. (2011). Some new sequence spaces derived by the domain of the triple band matrix. Computers and Mathematics with Applications, 62(2), 641-650. https://doi.org/10.1016/j.camwa.2011.05.045
• Sönmez, A. (2020). Some notes on the new sequence space b_p^(r,s) (D). Gazi University Journal of Science, 33(2), 476-490. https://doi.org/10.35378/gujs.522691
• Stieglitz, M., Tietz, H. (1977). Matrixtransformationen von folgenräumen. eine ergebnisübersicht. Mathematische Zeitschrift, 154, 1-16. https://doi.org/10.1007/BF01215107
• Wilansky, A. (1984). Summability Throught Functional Analysis (1), North-Holland Mathematics Studies vol 85, ISBN: 9780444557773, North-Holland, Amsterdam, 317.